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The big marbles in the first bag have a higher ratio of red to blue marbles than
dothesmallmarbles;thesameistrueabouttheratiointhesecondbag.But
considering all the marbles together, the small marbles have a higher ratio of
reds to blues than the big marbles do (in the combined bag: 70%
>
56%).
We argue that this marble example is a case of SP since it has the same math-
ematical structure as the type I version of SP. There are no causal assumptions
made in this example, no possible causal “confounding” and yet it seems para-
doxical. We believe this counter-example shows that at least sometimes, there is
a purely mathematical mistake about ratios that people customarily make. Some
causal theorists might be tempted to contend that even in this example there is
confounding between the effects of the marble size on the color with the effects
of the bag on the color. However, this confounding is not a causal confounding
since one
cannot
say that Bag 1
has caused
big marbles to become more likely
to be red or that Bag 2 has caused big marbles to become more likely to be
blue. In short, one must admit that the above counter-example does not involve
causal intuitions, yet it is still a case of SP.
5
“What-To-Do” Question and Causal Accounts
In the case of SP, “what-to-do” questions arise when investigators are confronted
with choosing between two conflicting statistics. For example, in Table 1, the
conflict is between the uncombined statistics of the two departments and their
combined statistics. Which one should they use to act? It is evident that many
interesting cases of choosing actions arise when we infer causes/patterns from
proportions. The standard examples
9
deal with cases in which “what-to-do”
questions become preeminent. But it should be clear in what follows that there
is no unique response to this sort of question for all cases of the paradox. Consider
Table 6 based on data about 80 patients. 40 patients were given the treatment,
T, and 40 assigned to a control,
∼
T. Patients either recovered, R, or didn't
recover,
∼
R. There were two types of patients, males (M) and females (
∼
M).
Tabl e 6.
Simpson's Paradox (Medical Example)
Recovery
Rates
M
∼
M
Two
Groups
Overall Recovery
Rates
R
∼
R
R
∼
R
M
∼
M
T
18
12
2
8
60%
20%
50%
∼
T
7
3
9
21
70%
30%
40%
One would think that treatment is preferable to control in the combined
statistics, whereas, given the statistics of the sub-population, one gathers the
impression that control is better for both men and women. Given a person of
unknown gender, would one recommend the control? The standard response is
clear: control is better for a person of unknown gender (since Pr(R
|∼
T)
>
9
These recommendations are standard because they are agreed upon by philosophers
[8], statisticians, and computer scientists [9].
